卷积神经网络（CNN）的量化是缓解CNN部署的计算负担，尤其是在低资源边缘设备上的常见方法。但是，对于神经网络所涉及的计算类型，固定点算术并不是自然的。在这项工作中，我们探索了使用基于PDE的观点和分析来改善量化CNN的方法。首先，我们利用总变化方法（电视）方法将边缘意识平滑应用于整个网络的特征图。这旨在减少值分布的异常值并促进零件恒定图，这更适合量化。其次，我们考虑用于图像分类的常见CNN的对称和稳定变体，以及用于图源分类的图形卷积网络（GCN）。我们通过几个实验证明，正向稳定性的性质保留了在不同量化速率下网络的作用。结果，稳定的量化网络的行为与非量化的网络相似，即使它们依赖于较少的参数。我们还发现，有时，稳定性甚至有助于提高准确性。对于敏感，资源受限，低功率或实时应用（例如自动驾驶），这些属性特别感兴趣。

在本文中，我们提出了解决稳定性和卷积神经网络（CNN）的稳定性和视野的问题的神经网络。作为提高网络深度或宽度以提高性能的替代方案，我们提出了与全球加权拉普拉斯，分数拉普拉斯和逆分数拉普拉斯算子有关的基于积分的空间非识别算子，其在物理科学中的几个问题中出现。这种网络的前向传播由部分积分微分方程（PIDE）启发。我们在自动驾驶中测试基准图像分类数据集和语义分段任务的提出神经架构的有效性。此外，我们调查了这些密集的运营商的额外计算成本以及提出神经网络的前向传播的稳定性。

While machine learning is traditionally a resource intensive task, embedded systems, autonomous navigation, and the vision of the Internet of Things fuel the interest in resource-efficient approaches. These approaches aim for a carefully chosen trade-off between performance and resource consumption in terms of computation and energy. The development of such approaches is among the major challenges in current machine learning research and key to ensure a smooth transition of machine learning technology from a scientific environment with virtually unlimited computing resources into everyday's applications. In this article, we provide an overview of the current state of the art of machine learning techniques facilitating these real-world requirements. In particular, we focus on deep neural networks (DNNs), the predominant machine learning models of the past decade. We give a comprehensive overview of the vast literature that can be mainly split into three non-mutually exclusive categories: (i) quantized neural networks, (ii) network pruning, and (iii) structural efficiency. These techniques can be applied during training or as post-processing, and they are widely used to reduce the computational demands in terms of memory footprint, inference speed, and energy efficiency. We also briefly discuss different concepts of embedded hardware for DNNs and their compatibility with machine learning techniques as well as potential for energy and latency reduction. We substantiate our discussion with experiments on well-known benchmark datasets using compression techniques (quantization, pruning) for a set of resource-constrained embedded systems, such as CPUs, GPUs and FPGAs. The obtained results highlight the difficulty of finding good trade-offs between resource efficiency and predictive performance.

These notes were compiled as lecture notes for a course developed and taught at the University of the Southern California. They should be accessible to a typical engineering graduate student with a strong background in Applied Mathematics. The main objective of these notes is to introduce a student who is familiar with concepts in linear algebra and partial differential equations to select topics in deep learning. These lecture notes exploit the strong connections between deep learning algorithms and the more conventional techniques of computational physics to achieve two goals. First, they use concepts from computational physics to develop an understanding of deep learning algorithms. Not surprisingly, many concepts in deep learning can be connected to similar concepts in computational physics, and one can utilize this connection to better understand these algorithms. Second, several novel deep learning algorithms can be used to solve challenging problems in computational physics. Thus, they offer someone who is interested in modeling a physical phenomena with a complementary set of tools.

我们提出了一种多移民通道（MGIC）方法，该方法可以解决参数数量相对于标准卷积神经网络（CNN）中的通道数的二次增长。因此，我们的方法解决了CNN中的冗余，这也被轻量级CNN的成功所揭示。轻巧的CNN可以达到与参数较少的标准CNN的可比精度。但是，权重的数量仍然随CNN的宽度四倍地缩放。我们的MGIC体系结构用MGIC对应物代替了每个CNN块，该块利用了小组大小的嵌套分组卷积的层次结构来解决此问题。因此，我们提出的架构相对于网络的宽度线性扩展，同时保留了通道的完整耦合，如标准CNN中。我们对图像分类，分割和点云分类进行的广泛实验表明，将此策略应用于Resnet和MobilenetV3等不同体系结构，可以减少参数的数量，同时获得相似或更好的准确性。

神经运营商最近成为设计神经网络形式的功能空间之间的解决方案映射的流行工具。不同地，从经典的科学机器学习方法，以固定分辨率为输入参数的单个实例学习参数，神经运算符近似PDE系列的解决方案图。尽管他们取得了成功，但是神经运营商的用途迄今为止仅限于相对浅的神经网络，并限制了学习隐藏的管理法律。在这项工作中，我们提出了一种新颖的非局部神经运营商，我们将其称为非本体内核网络（NKN），即独立的分辨率，其特征在于深度神经网络，并且能够处理各种任务，例如学习管理方程和分类图片。我们的NKN源于神经网络的解释，作为离散的非局部扩散反应方程，在无限层的极限中，相当于抛物线非局部方程，其稳定性通过非本种载体微积分分析。与整体形式的神经运算符相似允许NKN捕获特征空间中的远程依赖性，而节点到节点交互的持续处理使NKNS分辨率独立于NKNS分辨率。与神经杂物中的相似性，在非本体意义上重新解释，并且层之间的稳定网络动态允许NKN的最佳参数从浅到深网络中的概括。这一事实使得能够使用浅层初始化技术。我们的测试表明，NKNS在学习管理方程和图像分类任务中占据基线方法，并概括到不同的分辨率和深度。

Although weight and activation quantization is an effective approach for Deep Neural Network (DNN) compression and has a lot of potentials to increase inference speed leveraging bit-operations, there is still a noticeable gap in terms of prediction accuracy between the quantized model and the full-precision model. To address this gap, we propose to jointly train a quantized, bit-operation-compatible DNN and its associated quantizers, as opposed to using fixed, handcrafted quantization schemes such as uniform or logarithmic quantization. Our method for learning the quantizers applies to both network weights and activations with arbitrary-bit precision, and our quantizers are easy to train. The comprehensive experiments on CIFAR-10 and ImageNet datasets show that our method works consistently well for various network structures such as AlexNet, VGG-Net, GoogLeNet, ResNet, and DenseNet, surpassing previous quantization methods in terms of accuracy by an appreciable margin. Code available at https://github.com/Microsoft/LQ-Nets

模型量化已成为加速深度学习推理的不可或缺的技术。虽然研究人员继续推动量化算法的前沿，但是现有量化工作通常是不可否认的和不可推销的。这是因为研究人员不选择一致的训练管道并忽略硬件部署的要求。在这项工作中，我们提出了模型量化基准（MQBench），首次尝试评估，分析和基准模型量化算法的再现性和部署性。我们为实际部署选择多个不同的平台，包括CPU，GPU，ASIC，DSP，并在统一培训管道下评估广泛的最新量化算法。 MQBENCK就像一个连接算法和硬件的桥梁。我们进行全面的分析，并找到相当大的直观或反向直观的见解。通过对齐训练设置，我们发现现有的算法在传统的学术轨道上具有大致相同的性能。虽然用于硬件可部署量化，但有一个巨大的精度差距，仍然不稳定。令人惊讶的是，没有现有的算法在MQBench中赢得每一项挑战，我们希望这项工作能够激发未来的研究方向。

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics, and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than classical feed-forward neural networks, recurrent neural networks, and convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering, In this work, we review such methods and extend them for parametric studies as well as for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications.

混合精确的深神经网络达到了硬件部署所需的能源效率和吞吐量，尤其是在资源有限的情况下，而无需牺牲准确性。但是，不容易找到保留精度的最佳每层钻头精度，尤其是在创建巨大搜索空间的大量模型，数据集和量化技术中。为了解决这一困难，最近出现了一系列文献，并且已经提出了一些实现有希望的准确性结果的框架。在本文中，我们首先总结了文献中通常使用的量化技术。然后，我们对混合精液框架进行了彻底的调查，该调查是根据其优化技术进行分类的，例如增强学习和量化技术，例如确定性舍入。此外，讨论了每个框架的优势和缺点，我们在其中呈现并列。我们最终为未来的混合精液框架提供了指南。

We introduce a method to train Quantized Neural Networks (QNNs) -neural networks with extremely low precision (e.g., 1-bit) weights and activations, at run-time. At traintime the quantized weights and activations are used for computing the parameter gradients. During the forward pass, QNNs drastically reduce memory size and accesses, and replace most arithmetic operations with bit-wise operations. As a result, power consumption is expected to be drastically reduced. We trained QNNs over the MNIST, CIFAR-10, SVHN and ImageNet datasets. The resulting QNNs achieve prediction accuracy comparable to their 32-bit counterparts. For example, our quantized version of AlexNet with 1-bit weights and 2-bit activations achieves 51% top-1 accuracy. Moreover, we quantize the parameter gradients to 6-bits as well which enables gradients computation using only bit-wise operation. Quantized recurrent neural networks were tested over the Penn Treebank dataset, and achieved comparable accuracy as their 32-bit counterparts using only 4-bits. Last but not least, we programmed a binary matrix multiplication GPU kernel with which it is possible to run our MNIST QNN 7 times faster than with an unoptimized GPU kernel, without suffering any loss in classification accuracy. The QNN code is available online.

Although considerable progress has been obtained in neural network quantization for efficient inference, existing methods are not scalable to heterogeneous devices as one dedicated model needs to be trained, transmitted, and stored for one specific hardware setting, incurring considerable costs in model training and maintenance. In this paper, we study a new vertical-layered representation of neural network weights for encapsulating all quantized models into a single one. With this representation, we can theoretically achieve any precision network for on-demand service while only needing to train and maintain one model. To this end, we propose a simple once quantization-aware training (QAT) scheme for obtaining high-performance vertical-layered models. Our design incorporates a cascade downsampling mechanism which allows us to obtain multiple quantized networks from one full precision source model by progressively mapping the higher precision weights to their adjacent lower precision counterparts. Then, with networks of different bit-widths from one source model, multi-objective optimization is employed to train the shared source model weights such that they can be updated simultaneously, considering the performance of all networks. By doing this, the shared weights will be optimized to balance the performance of different quantized models, thus making the weights transferable among different bit widths. Experiments show that the proposed vertical-layered representation and developed once QAT scheme are effective in embodying multiple quantized networks into a single one and allow one-time training, and it delivers comparable performance as that of quantized models tailored to any specific bit-width. Code will be available.

模型二进制化是一种压缩神经网络并加速其推理过程的有效方法。但是，1位模型和32位模型之间仍然存在显着的性能差距。实证研究表明，二进制会导致前进和向后传播中的信息损失。我们提出了一个新颖的分布敏感信息保留网络（DIR-NET），该网络通过改善内部传播和引入外部表示，将信息保留在前后传播中。 DIR-NET主要取决于三个技术贡献：（1）最大化二进制（IMB）的信息：最小化信息损失和通过重量平衡和标准化同时同时使用权重/激活的二进制误差； （2）分布敏感的两阶段估计器（DTE）：通过共同考虑更新能力和准确的梯度来通过分配敏感的软近似来保留梯度的信息； （3）代表性二进制 - 意识蒸馏（RBD）：通过提炼完整精确和二元化网络之间的表示来保留表示信息。 DIR-NET从统一信息的角度研究了BNN的前进过程和后退过程，从而提供了对网络二进制机制的新见解。我们的DIR-NET中的三种技术具有多功能性和有效性，可以在各种结构中应用以改善BNN。关于图像分类和客观检测任务的综合实验表明，我们的DIR-NET始终优于主流和紧凑型体系结构（例如Resnet，vgg，vgg，EfficityNet，darts和mobilenet）下最新的二进制方法。此外，我们在现实世界中的资源有限设备上执行DIR-NET，该设备可实现11.1倍的存储空间和5.4倍的速度。

受生物神经元的启发，激活功能在许多现实世界中常用的任何人工神经网络的学习过程中起着重要作用。文献中已经提出了各种激活功能，用于分类和回归任务。在这项工作中，我们调查了过去已经使用的激活功能以及当前的最新功能。特别是，我们介绍了多年来激活功能的各种发展以及这些激活功能的优势以及缺点或局限性。我们还讨论了经典（固定）激活功能，包括整流器单元和自适应激活功能。除了基于表征的激活函数的分类法外，还提出了基于应用的激活函数的分类法。为此，对MNIST，CIFAR-10和CIFAR-100等分类数据集进行了各种固定和自适应激活函数的系统比较。近年来，已经出现了一个具有物理信息的机器学习框架，以解决与科学计算有关的问题。为此，我们还讨论了在物理知识的机器学习框架中使用的激活功能的各种要求。此外，使用Tensorflow，Pytorch和Jax等各种机器学习库之间进行了不同的固定和自适应激活函数进行各种比较。

由于神经网络变得更加强大，因此在现实世界中部署它们的愿望是一个上升的愿望;然而，神经网络的功率和准确性主要是由于它们的深度和复杂性，使得它们难以部署，尤其是在资源受限的设备中。最近出现了神经网络量化，以满足这种需求通过降低网络的精度来降低神经网络的大小和复杂性。具有较小和更简单的网络，可以在目标硬件的约束中运行神经网络。本文调查了在过去十年中开发的许多神经网络量化技术。基于该调查和神经网络量化技术的比较，我们提出了该地区的未来研究方向。

Deep neural networks provide unprecedented performance gains in many real world problems in signal and image processing. Despite these gains, future development and practical deployment of deep networks is hindered by their blackbox nature, i.e., lack of interpretability, and by the need for very large training sets. An emerging technique called algorithm unrolling or unfolding offers promise in eliminating these issues by providing a concrete and systematic connection between iterative algorithms that are used widely in signal processing and deep neural networks. Unrolling methods were first proposed to develop fast neural network approximations for sparse coding. More recently, this direction has attracted enormous attention and is rapidly growing both in theoretic investigations and practical applications. The growing popularity of unrolled deep networks is due in part to their potential in developing efficient, high-performance and yet interpretable network architectures from reasonable size training sets. In this article, we review algorithm unrolling for signal and image processing. We extensively cover popular techniques for algorithm unrolling in various domains of signal and image processing including imaging, vision and recognition, and speech processing. By reviewing previous works, we reveal the connections between iterative algorithms and neural networks and present recent theoretical results. Finally, we provide a discussion on current limitations of unrolling and suggest possible future research directions.

这本数字本书包含在物理模拟的背景下与深度学习相关的一切实际和全面的一切。尽可能多，所有主题都带有Jupyter笔记本的形式的动手代码示例，以便快速入门。除了标准的受监督学习的数据中，我们将看看物理丢失约束，更紧密耦合的学习算法，具有可微分的模拟，以及加强学习和不确定性建模。我们生活在令人兴奋的时期：这些方法具有从根本上改变计算机模拟可以实现的巨大潜力。

标准的神经网络可以近似一般的非线性操作员，要么通过数学运算符的组合（例如，在对流 - 扩散反应部分微分方程中）的组合，要么仅仅是黑匣子，例如黑匣子，例如一个系统系统。第一个神经操作员是基于严格的近似理论于2019年提出的深层操作员网络（DeepOnet）。从那时起，已经发布了其他一些较少的一般操作员，例如，基于图神经网络或傅立叶变换。对于黑匣子系统，对神经操作员的培训仅是数据驱动的，但是如果知道管理方程式可以在培训期间将其纳入损失功能，以开发物理知识的神经操作员。神经操作员可以用作设计问题，不确定性量化，自主系统以及几乎任何需要实时推断的应用程序中的代替代物。此外，通过将它们与相对轻的训练耦合，可以将独立的预训练deponets用作复杂多物理系统的组成部分。在这里，我们介绍了Deponet，傅立叶神经操作员和图神经操作员的评论，以及适当的扩展功能扩展，并突出显示它们在计算机械师中的各种应用中的实用性，包括多孔媒体，流体力学和固体机制， 。

神经网络的经典发展主要集中在有限维欧基德空间或有限组之间的学习映射。我们提出了神经网络的概括，以学习映射无限尺寸函数空间之间的运算符。我们通过一类线性积分运算符和非线性激活函数的组成制定运营商的近似，使得组合的操作员可以近似复杂的非线性运算符。我们证明了我们建筑的普遍近似定理。此外，我们介绍了四类运算符参数化：基于图形的运算符，低秩运算符，基于多极图形的运算符和傅里叶运算符，并描述了每个用于用每个计算的高效算法。所提出的神经运营商是决议不变的：它们在底层函数空间的不同离散化之间共享相同的网络参数，并且可以用于零击超分辨率。在数值上，与现有的基于机器学习的方法，达西流程和Navier-Stokes方程相比，所提出的模型显示出卓越的性能，而与传统的PDE求解器相比，与现有的基于机器学习的方法有关的基于机器学习的方法。

深度学习技术在各种任务中都表现出了出色的有效性，并且深度学习具有推进多种应用程序（包括在边缘计算中）的潜力，其中将深层模型部署在边缘设备上，以实现即时的数据处理和响应。一个关键的挑战是，虽然深层模型的应用通常会产生大量的内存和计算成本，但Edge设备通常只提供非常有限的存储和计算功能，这些功能可能会在各个设备之间差异很大。这些特征使得难以构建深度学习解决方案，以释放边缘设备的潜力，同时遵守其约束。应对这一挑战的一种有希望的方法是自动化有效的深度学习模型的设计，这些模型轻巧，仅需少量存储，并且仅产生低计算开销。该调查提供了针对边缘计算的深度学习模型设计自动化技术的全面覆盖。它提供了关键指标的概述和比较，这些指标通常用于量化模型在有效性，轻度和计算成本方面的水平。然后，该调查涵盖了深层设计自动化技术的三类最新技术：自动化神经体系结构搜索，自动化模型压缩以及联合自动化设计和压缩。最后，调查涵盖了未来研究的开放问题和方向。